Bernoulli

  • Bernoulli
  • Volume 24, Number 1 (2018), 672-698.

Curvature and transport inequalities for Markov chains in discrete spaces

Max Fathi and Yan Shu

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Abstract

We study various transport-information inequalities under three different notions of Ricci curvature in the discrete setting: the curvature-dimension condition of Bakry and Émery (In Séminaire de Probabilités, XIX, 1983/84 (1985) 177–206 Springer), the exponential curvature-dimension condition of Bauer et al. (Li-Yau Inequality on Graphs (2013)) and the coarse Ricci curvature of Ollivier (J. Funct. Anal. 256 (2009) 810–864). We prove that under a curvature-dimension condition or coarse Ricci curvature condition, an $L_{1}$ transport-information inequality holds; while under an exponential curvature-dimension condition, some weak-transport information inequalities hold. As an application, we establish a Bonnet–Myers theorem under the curvature-dimension condition $\operatorname{CD}(\kappa,\infty)$ of Bakry and Émery (In Séminaire de Probabilités, XIX, 1983/84 (1985) 177–206 Springer).

Article information

Source
Bernoulli, Volume 24, Number 1 (2018), 672-698.

Dates
Received: October 2015
Revised: July 2016
First available in Project Euclid: 27 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.bj/1501142459

Digital Object Identifier
doi:10.3150/16-BEJ892

Mathematical Reviews number (MathSciNet)
MR3706773

Zentralblatt MATH identifier
06778344

Keywords
curvature discrete spaces functional inequalities Markov chains optimal transport

Citation

Fathi, Max; Shu, Yan. Curvature and transport inequalities for Markov chains in discrete spaces. Bernoulli 24 (2018), no. 1, 672--698. doi:10.3150/16-BEJ892. https://projecteuclid.org/euclid.bj/1501142459


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